Question

Statement: “Where there is a cloud, there is a rain.” Which of the following statements, if true, would show that the above statement is false?

• A. Sometimes there is cloud, but there is no rain.
• B. Sometimes there is rain, but there is no cloud.
• C. There is no rain where there is no cloud.
• D. of the above.

Hint:

To falsify “If $P$ then $Q$,” look for $P$ true but $Q$ false — that’s the only scenario that disproves it.

Answer 1

The Correct Option is A

The statement “Where there is a cloud, there is a rain” means: If cloud exists, then rain exists. Symbolically: $C \Rightarrow R$.
To prove this false, we need a case where $C$ is true but $R$ is false — that is, there are clouds but no rain.
Option (a) exactly gives this counterexample.
Option (b) does not contradict the original statement because it talks about rain without clouds, which was not restricted by the original statement.
Option (c) is actually the converse of the contrapositive and does not contradict the given statement.
Thus, only option (a) shows the original statement to be false.